$D\to \pi ,l\nu$ Semileptonic Decays, $\mathrm{\mid }{V}_{cd}\mathrm{\mid }$ and 2${}^{nd}$ Row Unitarity from Lattice QCD

We present a new calculation of the $D\to \pi ,l\nu$ semileptonic form factor $f_{+}^{D\to \pi }(q^2)$ at $q^2=0$ based on HISQ charm and light valence quarks on MILC $N_f=2+1$ lattices. Using methods developed recently for HPQCD’s study of $D\to K,l\nu$ decays, we find $f^{D \rightarrow \pi}+(0) = 0.666(29)$. This signifies a better than factor of two improvement in errors for this quantity compared to previous calculations. Combining the new result with CLEO-c branching fraction data, we extract the CKM matrix element $\mathrm{\mid }{V}_{cd}\mathrm{\mid }=0.225(6{)}_{exp\mathrm{.}}(10{)}_{lat\mathrm{.}}$, where the first error comes from experiment and the second from theory. With a total error of $\sim 5.3$% the accuracy of direct determination of $\mathrm{\mid }{V}_{cd}\mathrm{\mid }$ from $D$ semileptonic decays has become comparable to (and in good agreement with) that from neutrino scattering. We also check for second row unitarity using this new $\mathrm{\mid }{V}_{cd}\mathrm{\mid }$, HPQCD’s earlier $\mathrm{\mid }{V}_{cs}\mathrm{\mid }$ and $\mathrm{\mid }{V}_{cb}\mathrm{\mid }$ from the Fermilab Lattice & MILC collaborations. We find $|V{cd}|^2 + |V_{cs}|^2 + |V_{cb}|^2 = 0.976(50)$, improving on the current PDG2010 value.